Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges.

For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex embeddable in $\mathbb{R}^d$ has an embedding which is linear on every face?

It is true when $d \ge 2k+1$ by putting things in general position.

I am especially interested to know if anything is known about the case $(k,d)=(2,3)$.

I vaguely remember an old conjecture of Branko Grünbaum that every triangulation of the torus admitting a "straight" embedding in $\mathbb{R}^3$ but I don't know a reference (or whether this is still open).